The following expands upon the comment I made earlier now that I've searched a bit through Igros Moshe.
Here is the original wording of the teshuva:
אבל פשוט שהוא רק כשלמדו כפי פירוש אמת שהוא כאחד מפירושי רבותינו
הראשונים ולא בפירושים מאנשים דעלמא שרובן בדואים ואינם אמת והרבה יש גם
כעין מגלה פנים בתורה שלא כהלכה
Translation:
But it's simple that this is only when they learned according to a
true commentary which is like one of the commentaries of our rabbis
the Rishonim, and not with commentaries from ordinary people which are
mostly made up and aren't truthful and many possess a form of megaleh
panim baTorah shelo k'halacha.
This is to say that Rav Moshe wasn't limiting the ability to make a siyum on sifrei Tanach only to one who learned Tanach along with a commentary of the Rishonim, but also allowed room for one who studied Tanach along with a commentary that was "like" those written by the classic rishonim and were in the same spirit, as opposed the commentaries of a "plain old person" (likely with the maskillic tendencies that were prevalent within the 100 or so years prior to the teshuva).
So Rav Moshe allowed for meforshim that were more contemporary, but clung to the spirit of traditional interpretations that didn't veer away from the Torah and Halacha. In this way, commentaries like the Malbim, the Netziv's HaAmek Davar and some others (even before scanning through Rav Moshe's seforim) would seem to be in the like the Rishonim group vs. in the plain/maskillic/non-halachic group.
Upon review we actually can find the following places in Igros Moshe where Rav Moshe respectfully mentions the perush of the Malbim with honor:
- In Chelek Kodshim v'Taharos printed at the end of Orach Chaim vol
1, in siman 2 he mentions a support from פירוש המלבי"ם
- In Yoreh Deah vol 1, siman 192 he quotes Malbim's peirush throughout
- In Orach Chaim vol 5, siman 38.7 he quotes the Malbim
Additionally, Rav Moshe treats the Netziv's העמק דבר with honor/respect:
- In Chelek Kodshim v'Taharos printed at the end of Orach Chaim vol
1, toward the end of siman 10, he mentions a question on: ומה שהובא...בשם הגאון הנצי"ב בספרו העמק דבר
- He also mentions the העמק דבר respectfully in Orach Chaim 3:69 and in Yoreh Deah 4:38.3
Final note - if one wants to flesh this out and get a fuller picture, it shouldn't be so hard to find phone numbers and email addresses for members of Rav Moshe's family and talmidim who had direct contact with Rav Moshe zt"l and likely heard his opinions on these matters in-person.